\section{Hybrid Modeling}

\frame {
  \frametitle{Outline}
  \tableofcontents[sectionstyle=show/shaded,subsectionstyle=hide]
}

\subsection{Hybrid Modeling}

%\frame {
%  \frametitle{Central Modeling Paradigm}
%  \begin{misc}
%    \em Bipedal walking can be modeleded using rigid chains with points which are %periodically in fixed contact with the ground.
%  \end{misc}
%  \begin{figure}
%    \centering
%    \includegraphics[height=.65\textheight]{domainbreaking}
%  \end{figure}
%}

\frame {
  \frametitle{Hybrid Systems}
  \vspace{-.3cm}
  \begin{columns}[t]
    \begin{column}{.63\textwidth}
      \vspace{-.3cm}
      \begin{definition}
        A \alert{hybrid control system} is a tuple \vspace{-.3cm}
        $$\HCS = \hcsystem, \vspace{-.4cm}$$
      where
      \begin{itemize}
      \item
        $\Domain \subset \mathcal{X}$ is the {\em domain of admissiblity} with state space $\mathcal{X}$,
      \item
        $\ControlSet$ is a set of {\em admissible controls},
      \item
        $\Guard$ is a {\em guard} or {\em switching surface},
      \item
        $\ResetMap$ is a smooth map called the {\em reset map}
      \item
        $(f, g)$ is a control system on $\Domain$, i.e., $\dot{x} = f(x) + g(x) \, u$.
      \end{itemize}
      A \alert{hybrid system} $\HS = \hsystem$ is a hybrid control system with $\ControlSet = \{0\}$ so $\dot{x} = f(x)$.
      \end{definition}
    \end{column}
    \begin{column}{.4\textwidth}
      \begin{figure}
        \centering
        \includegraphics[width=.9\textwidth]{hsystem.pdf}\\
      \end{figure}
      A \textcolor{blue}{simple hybrid system}:\vspace{-.3cm}
      $$\HS = \hsystem$$
    \end{column}
  \end{columns}
}

\frame{
  \frametitle{Robotic Model}
  \begin{columns}
    \begin{column}{.35\textwidth}
      \begin{figure}
        \centering
        \caption{Configuration of the bipedal model:}
        \subfloat{\includegraphics[height=4cm]{robot_config}}
      \end{figure}
    \end{column}
    \begin{column}{.6\textwidth}
      \begin{figure}
        \centering
        \caption{Based on the robot AMBER:}
        \subfloat{\includegraphics[height=4cm]{amber_robot}}
        \subfloat{\includegraphics[height=4cm]{amber_solidworks}}
      \end{figure}
    \end{column}
  \end{columns}
}

\frame{
  \frametitle{Nonlinear Model}
    \begin{columns}
     \begin{column}{.65\textwidth}
        For coordinates $(\q^{T}, \dq^{T})^{T} \in T\ConfigurationSpace = T\mathcal{S}$, the dynamics satisfies
        \begin{align*}
          \D(\q) \, \ddq + \C(\q, \dq) \, \dq + \G(\q) = \xB(\q) \, \q
        \end{align*}
        which can also be written
        \begin{align*}
          f(\q, \dq) &= \left(\begin{array}{c}
            \dq\\
            \D^{-1}(\q) (-\C(\q, \dq) \, \dq - \G(\q))
          \end{array}\right),\\
          g(\q) &= \left(\begin{array}{c}
            \mathbf{0}_{m \times m}\\
            \D^{-1}(\q) \xB(\q)
          \end{array}\right).
      \end{align*}
    \end{column}
    \begin{column}{.35\textwidth}
      \begin{figure}
        \centering
        \caption{Coordinates of biped:}
        \subfloat{\includegraphics[height=5cm]{robot_angles}}
      \end{figure}
    \end{column}
  \end{columns}
}

\frame[t] {
  \frametitle{Impact Model}
  Introduce extended coordinates $\qe = (p_{x}, p_{y}, \q^{T})^{T} \in \R^{2} \times \ConfigurationSpace = \ConfigurationSpace_{e}$. Angular momentum balance based on H{\"u}rm{\"u}zl{\"u} and Marghitu:
  \begin{assumptions}
    \begin{itemize}
      \item Rigid-body plastic impacts
      \item Enough friction to prevent slipping
      \item Impact points do not rebound
      \item Motors do not produce impulses
      \item No instantaneous change in configuration, i.e., $\qe^{-} = \qe^{+}$
    \end{itemize}
  \end{assumptions}
  Under these assumptions, the short-scale dynamics satisfies
  \begin{align*}
    \left[\begin{array}{c c}
        \D(\qe) & -\Jacobian^{T}(\qe)\\
        \Jacobian(\qe) & \mathbf{0}_{2 \times 2}
      \end{array}\right]
    \left[\begin{array}{c}
        \dq^{+}\\
        \delta F(\qe, \dqe)
      \end{array}\right]
    = \left[\begin{array}{c c}
        \D(\qe) \, \dqe^{-}\\
        \mathbf{0}_{2}
      \end{array}\right]
\end{align*}
}

\frame[t] {
  \frametitle{Some Concepts Applied to Hybrid Systems}
  \newtheorem{stability}{Stability}
  \newtheorem{robustness}{Robustness}
  \newtheorem{hlw}{Humanlike Walking}
  \begin{stability}
    \alert{Poincar\'e Analysis}: Construct hyperplane transverse to the orbit and analyze the return map.
  \end{stability}
  
  \vspace{3mm}
    
  \begin{robustness}
    \alert{For Bipeds}: The ability to walk over rough terrain without falling.
  \end{robustness}
  
  \vspace{3mm}
  
  \begin{hlw}
    \alert{Gait Analysis}: Do the desired trajectories fall within one standard deviation of healthy human walking?
  \end{hlw}
}